Which is a recursive formula for the sequence 99.4, 0, –99.4, –198.8, where f(1) = 99.4?

f(n + 1) = f(n) + 99.4, n ≥ 1
f(n + 1) = f(n) – 99.4, n ≥ 1
f(n + 1) = 99.4f(n), n ≥ 1
f(n + 1) = –99.4f(n), n ≥ 1

Let's analyze the given sequence:  
- \( f(1) = 99.4 \)
- \( f(2) = 0 \)
- \( f(3) = -99.4 \)
- \( f(4) = -198.8 \)

Each term seems to be decreasing by a constant value, suggesting a recursive relationship that involves subtraction or addition. Let's check each formula:

1. **\( f(n+1) = f(n) + 99.4 \)**:  
   This would imply each term is increasing by 99.4, which is not the case.

2. **\( f(n+1) = f(n) - 99.4 \)**:  
   Let's check:
   - \( f(2) = f(1) - 99.4 = 99.4 - 99.4 = 0 \)
   - \( f(3) = f(2) - 99.4 = 0 - 99.4 = -99.4 \)
   - \( f(4) = f(3) - 99.4 = -99.4 - 99.4 = -198.8 \)

   This matches the sequence perfectly.

3. **\( f(n+1) = 99.4f(n) \)**:  
   This would imply multiplying each term by 99.4, which doesn't match the given sequence.

4. **\( f(n+1) = -99.4f(n) \)**:  
   This would imply multiplying by -99.4 each time, which also doesn't fit the pattern.

The correct recursive formula is:
\[ f(n+1) = f(n) - 99.4 \quad \text{for} \quad n \geq 1 \]

Would you like more details or have further questions?

Here are some related questions to deepen understanding:
1. How does the sequence behave if we modify the initial value \( f(1) \)?
2. What is the explicit (non-recursive) formula for this sequence?
3. How does the difference in a sequence relate to its recursive formula?
4. Can you apply this type of recursive formula to geometric sequences?
5. What are the properties of arithmetic sequences like this one?

**Tip**: In a recursive formula, always check how the terms progress by calculating a few values from the given sequence.

Which is a recursive formula for the sequence 99.4, 0, –99.4, –198.8, where f(1) = 99.4?

Explore how to find the recursive formula for the sequence 99.4, 0, –99.4, –198.8, where f(1) = 99.4. This problem demonstrates the use of arithmetic sequences and recursive formulas. Perfect for students in Grades 9-12 looking to understand recursive patterns in sequences.

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